where , and is a domain in . We introduce the novel notion of symmetric systems. The above system is said to be symmetric if the matrix of gradient of all components of is symmetric. It seems that this concept is crucial to prove Liouville theorems, when , and regularity results, when , for stable solutions of the above system for a general nonlinearity . Moreover, we provide an improvement for a linear Liouville theorem given by Fazly and Ghoussoub in 2013 that is a key tool to establish De Giorgi type results in lower dimensions for elliptic equations and systems.

[1]
Giovanni Alberti, Luigi Ambrosio, and Xavier Cabré, On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math. 65 (2001), no. 1-3, 9-33. Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday. MR 1843784 (2002f:35080), https://doi.org/10.1023/A:1010602715526

Giovanni Alberti, Luigi Ambrosio, and Xavier Cabré, On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math. 65 (2001), no. 1-3, 9-33. Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday. MR 1843784 (2002f:35080), https://doi.org/10.1023/A:1010602715526